They're both unbiased. They are both sample means. But the first one has more observations hence it has a smaller variance. Thus it is preferable.

Let m=n and also 2n.

IN BOTH cases. So they are both unbiased.

NOW using independence we can likewise obtain

.

Hence the winner is the one with a larger sample size, which is the first one.

AND, there is no need for normality here whatsoever.